The Chern-Simons number as a dynamical variable

In the standard electroweak theory that describes nature, the Chern-Simons number associated with the vacua as well as the unstable sphaleron solutions play a crucial role in the baryon number violating processes. We recall why the Chern-Simons number should be generalized from a set of discrete values to a dynamical (quantum) variable. Via the construction of an appropriate Hopf invariant and the winding number, we discuss how the geometric information in the gauge fields is also captured in the Higgs field. We then discuss the choice of the Hopf variable in relation to the Chern-Simons variable.